Nehari and Nevanlinna-Pick problems and holomorphic extensions in the polydisk in terms of restricted BMO
Let \(T^ d\), \(d \geq 1\) be the \(d\)-dimensional torus, \(P\) be the class of trigonometric polynomials in \(T^ d\); \(P_ 1 = \{f \in P : \hat f(m_ 1, \dots, m_ d) = 0\), if \(m_ k < 0\) for some \(k = 1, \dots, d\}\), \(P_ 2 = \{f \in P : f \perp P_ 1\) in \(L^ 2(T^ d)\}\); \(H^ 2\) the closure of \(P_ 1\) in \(L^ 2\) and \(H^{2 \perp}\) the closure of \(P_ 2\). A (big) Hankel form in \(T^ d\), \(d \geq 1\), is a sesquilinear form: \(B : P_ 1 \times P_ 2 \to C\) such that \(B(f,g) = L(f \overline g)\), for some linear form \(L : P \to C\). Particularly, if \(\varphi \in L^ 2\) then \(B_ \varphi (f,g) = \int f \overline g \varphi\) is a Hankel form in \(T^ d\). The authors define the BMOr space (BMO restricted space, \(L^ \infty \subseteq \text{BMO}r \subseteq \text{BMO})\) in order to obtain the following \(d\)-dimensional variant of Nehari's theorem. a) A (big) Hankel form \(B\) in \(T^ d\), \(d \geq 1\) is bounded if there exists \(\varphi \in \text{BMOr} (T^ d) \cap H^{2 \perp} (T^ d)\) such that \(B = B_ \varphi\) and \(\| \varphi \|_{\text{BMOr}} \leq \| B \|\). b) Conversely, if \(\varphi \in \text{BMOr} (T^ d)\) then \(B = B_ \varphi\) is bounded and \(\| B \| \leq \sqrt d \| \varphi \|_{\text{BMOr}}\). Using this result they solve Nevanlinna-Pick type interpolation problems or holomorphic extension problems in the polydisk.
scientific article; zbMATH DE number 624769